Local bi-Lipschitz classification of semialgebraic surfaces

TL;DR

This paper develops bi-Lipschitz invariants for finitely determined map germs to classify their local and link structures, providing new criteria for bi-Lipschitz equivalence in real and complex settings.

Contribution

It introduces new invariants for finitely determined map germs and establishes conditions under which the bi-Lipschitz type of a germ determines its link and double point set.

Findings

Positive answer for the first question when 2n-1 ≤ p.

Complete invariants for the bi-Lipschitz classification in the case of maps from R^2 to R^3.

In the homogeneous case, the double point set's image is not needed for classification.

Abstract

We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ determine the bi-Lipschitz type of the link of $f$ and of the double point set of $f$? Reciprocally, given a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$, do the bi-Lipschitz types of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$? We provide a positive answer to the first question in the case of a finitely determined map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ where $2n-1 \leq p$ (Theorem 3.3). With regard to the second question, for a finitely determined map germ $f : (\mathbb{R}^2,0) \to (\mathbb{R}^3,0),$ we show that a complete set of invariants for the bi-Lipschitz classification with respect to the inner metric of $X_f=f(U)$, where $U$ is a small neighbourhood of the origin in $\mathbb R^2$, is is given by the link of $f$, the image of the double point set of $f$ and the polar curve of a generic projection into the plane (Proposition 4.13). In particular, in the homogeneous parametrization case $f: (\mathbb{R}^2, 0) \to (\mathbb{R}^3, 0)$ of corank 1, we do not need the hypothesis on the equivalence of the image of the double point set (Theorem 5.2). Finally, we apply our results to relate the $C^{0}- \mathcal A$ classes of finitely determined map germs $f$ of corank 1 with homogeneous parametrization and the inner bi-Lipschitz type of $X_f$ (Proposition 5.4).

Equations

$$({\mathbb{K}}^n,0)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\{(x,y)\in U\times U:f(x)=f(y),x\ne y\},$$

$$\tilde{D}(f):=\{(x,y)\in U\times U:\exists (x^{\prime },y^{\prime })\ne (x,y)\text{ such that }f(x,y)=f(x^{\prime },y^{\prime })\}\cup \Sigma (f),$$

$$\mathcal{L}(f)={\pi }^{-1}(t)\cap B_{\delta }$$

$$(f{|}_{{\tilde{S}}_{ϵ}^{n-1}})\times i{d}_{(0,ϵ)}:{\tilde{S}}_{ϵ}^{n-1}\times (0,ϵ)\to S_{ϵ}^{p-1}\times (0,ϵ)$$

$$C(f{|}_{{\tilde{S}}_{ϵ}^{n-1}}):{\tilde{S}}_{ϵ}^{n-1}\times [0,ϵ)/{\tilde{S}}_{ϵ}^{n-1}\times \{0\}\to S_{ϵ}^{p-1}\times [0,ϵ)/S_{ϵ}^{p-1}\times \{0\}$$

$$h:({\mathbb{K}}^n,0)\to ({\mathbb{K}}^n,0),k:({\mathbb{K}}^p,0)\to ({\mathbb{K}}^p,0)$$

$$k(f(x))=g(h(x)),k(f(x^{\prime }))=g(h(x^{\prime })).$$

$$f{|}_{U\setminus \{0\}}:U\setminus \{0\}\to W$$

$$f{|}_{S_{ϵ}^{n-1}}:S_{ϵ}^{n-1}\to S_{ϵ}^{p-1}$$

$$f{|}_{f^{-1}(S_{ϵ}^{p-1})}:f^{-1}(S_{ϵ}^{p-1})\to S_{ϵ}^{p-1}\text{ and }g{|}_{g^{-1}(S_{ϵ}^{p-1})}:g^{-1}(S_{ϵ}^{p-1})\to S_{ϵ}^{p-1}$$

$$d_{\mathrm{out}}(x,y)=\Vert x-y\Vert$$

$$d_{\mathrm{in}}(x,y)=\underset{\lambda \in \Lambda (x,y)}{\inf }l(\lambda )$$

$$d_{\mathrm{in}}(x,y)\le k{d}_{\mathrm{out}}(x,y).$$

$$K(A,B)=or{d}_r\left(\mathrm{dist}(A\cap S(0,r),B\cap S(0,r))\right),$$

$$T_{\beta }=\{(x,y)\in {\mathbb{R}}^2:0\le y\le x^{\beta },0\le x\le 1\}.$$

$$K(X_i,X_j)=K(Y_{{\sigma }_i},Y_{{\sigma }_j})i,j\in \{1,2,\dots ,l\}.$$

$$f(x,y)=(x,f_2(x,y),f_3(x,y)),g(x,y)=(x,g_2(x,y),g_3(x,y))$$

$$f(x,y)=(x,f_2(x,y),f_3(x,y)).$$

$$\frac{f_2(x,y)-f_2(x,y^{\prime })}{y-y^{\prime }}=0,\frac{f_3(x,y)-f_3(x,y^{\prime })}{y-y^{\prime }}=0.$$

$$f({\Gamma }_i)=(x,f_2(x,{\alpha }_{i}x),f_3(x,{\alpha }_{i}x)).$$

$$f(x,y):=(x,p(x,y),q(x,y)),$$

$${\mathbb{R}}_{+}^{\ast }\times X\to X,(t,(x_1,x_2,x_3))\mapsto (t{x}_1,t{x}_2,t{x}_3).$$

$$H(p)=O(H_L(p^{\prime }))\cap S_{ϵ}^2.$$

$$H(q)=O(H_L(q^{\prime }))\cap S_{{ϵ}^{\prime }}^2$$

$$d(H(p),H(q))\le d(H(p),w)+d(w,H(q)).$$

$$p^{\prime }=\frac{1}{ϵ}(a_1,a_2,a_3)=\frac{1}{ϵ}p.$$

$$w=\frac{{ϵ}^{\prime }}{R}(b_1,b_2,b_3)=\frac{{ϵ}^{\prime }}{R}{H}_L(p^{\prime }).$$

$$q^{\prime }=\frac{1}{{ϵ}^{\prime }}(x_1,x_2,x_3)=\frac{1}{{ϵ}^{\prime }}q,$$

$$H(q)=\frac{{ϵ}^{\prime }}{R}(y_1,y_2,y_3)=\frac{{ϵ}^{\prime }}{R}{H}_L(q^{\prime }).$$

Full text

Does the bi-Lipschitz type of a map germ $f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$ determine the bi-Lipschitz type of the link of $f$ and of the double point set of $f$? Reciprocally, given a map germ $f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$, do the bi-Lipschitz types of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the germ $f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$?

We provide a positive answer to the first question in the case of a finitely determined map germ $f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$ where $2n-1\leq p$ (Theorem 3.3).

With regard to the second question, for a finitely determined map germ $f:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0),$ we show (Proposition 4.12) that the bi-Lipschitz type of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the image $X_{f}=f(U),$ where $U$ is a small neighbourhood of the origin in $\mathbb{R}^{2}.$ In the case of a finitely determined map germ $f:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ of corank 1 with homogeneous parametrization, the bi-Lipschitz type of the link of $f$ determines the bi-Lipschitz type of the map germ (Theorem 5.2). Finally, we discuss the bi-Lipschitz equivalence of homogeneous surfaces in $\mathbb{R}^{3}$ (Theorem 6.1).

1. INTRODUCTION

Two map germs $f,g:(\mathbb{K}^{n},0)\to(\mathbb{K}^{p},0)$, where $\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$, are $\mathcal{A}$-equivalent if there exist diffeomorphism map germs $h:(\mathbb{K}^{n},0)\to(\mathbb{K}^{n},0)$ and $k:(\mathbb{K}^{p},0)\to(\mathbb{K}^{p},0)$ such that the following diagram is commutative

[TABLE]

that means $g=k\circ f\circ h^{-1}$. When $h$ and $k$ are homeomorphisms, then $f$ and $g$ are called topologically $\mathcal{A}$-equivalent, or ${\mathcal{C}^{0}}$-$\mathcal{A}$-equivalent. Two map germs $f$ and $g$ are bi-Lipschitz ${\mathcal{A}}$-equivalent if $h$ and $k$ in the above diagram are bi-Lipschitz homeomorphisms.

Note that bi-Lipschitz equivalence implies topological equivalence.

A map germ $f:(\mathbb{K}^{n},0)\to(\mathbb{K}^{p},0)$ is ${\mathcal{A}}$-finitely determined if there exists a positive integer $r$ such that for any $g$ with equality of the $r$-jets $j^{r}g(0)=j^{r}f(0)$, then $f$ and $g$ are $\mathcal{A}$-equivalent. In this case, we say that $f$ is $r$-${\mathcal{A}}$-finitely determined.

In this paper, we will consider only ${\mathcal{A}}$ equivalence classes and we will omit the symbol ${\mathcal{A}}$. Moreover, we consider only the case $n\leq p$.

The natural definition of double point set of $f$, that we will consider in this paper, denoted by $\widetilde{D}(f)$, is geometrically, the closure in $U\times U\subset\mathbb{K}^{n}\times\mathbb{K}^{n}$ of the set

[TABLE]

where $U$ is a sufficiently small neighbourhood of [math]. We denote by $D(f)=p_{1}(\widetilde{D}(f))$ the projection on the first factor via $p_{1}:\mathbb{K}^{n}\times\mathbb{K}^{n}\to\mathbb{K}^{n}$.

An alternative definition of double point set has been provided by D. Mond [18] in terms of ideals. It has been used by various authors, for example [20, 15, 16, 24]. It can be written as:

[TABLE]

where $\Sigma(f)$ is the singular set of $f$. Moreover when $\Sigma(f)$ is an isolated singularity, this definition coincides with the previous geometrical one.

Consider a representative of a finitely determined map germ $f:U\to(\mathbb{K}^{p},0)$, where $U\subset\mathbb{K}^{n}$ and $n\leq p.$ Let $X_{f}$ denote the image $f(U)$ of $f$. The intersection of $X_{f}$ with a sufficiently small sphere $S_{\epsilon}^{m}$ centered at the origin in $\mathbb{K}^{p}$ is called the link of the map germ $f$, and denoted by $link(f)$. Here, $m=p-1$ in the real case and $m=2p-1$ in the complex situation.

In the complex case, M. Goresky and R. MacPherson defined the complex link as follows:

We assume that the image $f(U)$ is Whitney stratified with complex analytic strata and we denote by $S$ the stratum containing the singular point [math]. We choose a complex analytic manifold $N$ meeting $S$ transversaly at [math] and a generic projection $\pi:N\cap f(U)\to\mathbb{C}$ sending $0\in\mathbb{C}^{p}$ to $0\in\mathbb{C}$. For $0<\varepsilon\ll\delta\ll 1$ the complex link of $S$ is defined by

[TABLE]

where $0<|t|<\varepsilon$ and $B_{\delta}$ is the intersection with $f(U)$ of the ball of radius $\delta$ centered at $0\in\mathbb{C}^{p}$.

The following conjecture can be proposed in various ways:

** Conjecture 1.1****.**

Let $f,g:(\mathbb{K}^{n},0)\to(\mathbb{K}^{p},0)$, where $2n-1\leq p$, be two finitely determined map germs. Then $f$ and $g$ are bi-Lipschitz equivalent if and only if the links $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and the images of the double point sets $f(D(f))$ and $g(D(g))$ are bi-Lipschitz equivalent.

In the complex situation, the considered link can be either the (real) link $link(f)$ or the complex link ${\mathcal{L}}(f).$

According to Whitney [28], when $2n-1\leq p$ and $n\geq 2$, the singularities of $f$ are isolated. Moreover, when $2n-1=p$ and $f$ is finitely determined, the image of the set of double points of $f$ is a curve, which can be embedded in the target space $\mathbb{K}^{2n-1}$ [13, 28]. Note that, according to [9], the dimensions $(n,p)$ satisfying $2n-1\leq p$ are nice dimensions in the sense of Mather (see for example [9, Table 1, page 181]).

Using equisingularity in family and in the complex case $\mathbb{K}=\mathbb{C}$, a counter-example to the conjecture has been provided by O. N. Silva [24, 25]. In his example the real links are topologically equivalent and the double point sets are bi-Lipschitz equivalent, but the germs are not bi-Lipschitz equivalent.

In the complex case, it would be interesting to characterize classes of maps and hypothesis for which the conjecture is true. The conjecture remains in the real situation. In this paper, we provide some elements for solving the “real” conjecture.

The paper has four parts:

I) The first part presented in the section 3 provides some bi-Lipschitz invariants of the double point sets and of the real links for $\mathcal{A}$-finitely determined map germs $f:(\mathbb{K}^{n},0)\to(\mathbb{K}^{p},0)$ for the general case $n\leq p$ and for the case $2n-1\leq p$. In Theorem 3.3 we show that the necessary conditions for Conjecture 1.1 are verified in the real case when $2n-1\leq p.$

II) In the second part, presented in the section 4, we show that for a finitely determined map germ $f:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0),$ the bi-Lipschitz type of the link of $f$ and of the double point set of $f$ determines the bi-Lipschitz type of $X_{f}$ with respect to the inner metric (see Definition 4.1). For this weaker equivalence relation, the proposed problem is completely solved (Proposition 4.12). The main tools used in this section are the Birbrair construction of Hölder Complexes (Definitions 4.4 and 4.8) and Birbrair Classification Theorem 4.11.

III) The third part is presented in the section 5. We prove that for a corank 1 finitely determined map germ with a homogeneous parametrization $f:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0),$ the bi-Lipschitz type of the link of $f$ determines the bi-Lipschitz type of $f$ (Theorem 5.2). This result provides a class of quasi-homogeneous surfaces for which the bi-Lipschitz type of the link determines the bi-Lipschitz type of the surface.

IV) Finally, in the fourth part presented in section 6, we consider the simpler problem of bi-Lipschitz equivalence of homogeneous surfaces in $\mathbb{R}^{3}$. In this case, we also prove that the bi-Lipschitz type of the link determines the bi-Lipschitz type of the surface. The homogeneous surfaces we consider in this section do not have a parametrization in general. More precisely, let $X\subset(\mathbb{R}^{3},0)$ and $Y\subset(\mathbb{R}^{3},0)$ be algebraic surfaces such that $X=\phi^{-1}(0)$ and $Y=\phi^{\prime-1}(0)$, where $\phi,\phi^{\prime}:\mathbb{R}^{3}\to\mathbb{R}$ are two homogeneous polynomials. We prove that if $link(X)$ and $link(Y)$ are bi-Lipschitz equivalent, then $X$ and $Y$ are bi-Lipschitz equivalent (Theorem 6.1).

Results on the related problem of $C^{0}$-$\mathcal{A}$-equivalence of finitely determined germs were given by J. Nuño Ballesteros and W. Marar in [15] (case $n=2,p=3$) and by R. Mendes and J. Nuño Ballesteros in [22] (case $n=2$ and $p=4$.)

2. Some well-known results

We present in this section some well-known results that will be used in the next sections of the paper.

Theorem 2.1** (Mather-Gaffney Geometric criterion, [27]).**

A map germ $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ is finitely determined if and only if, for every representative $f$ of the map germ, there exist a neighbourhood $U$ of 0 in $\mathbb{C}^{n}$ and a neighbourhood $V$ of 0 in $\mathbb{C}^{p}$, with $f(U)\subset V$, such that for all $y\in V\setminus\{0\}$, the set $S=f^{-1}(y)\cap\Sigma(f)$ is finite and $f:(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},y)$ is stable, where $\Sigma(f)$ is the set of critical points of $f$.

Let $f:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a finitely determined map germ. From the classical result of Whitney [28], we know that the stable singularities in these dimensions are transverse double points, triple points and cross-caps. In this case, Theorem 2.1 says that $f:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ is ${\mathcal{A}}$-finitely determined if and only if for every representative $f$, there exists a neighbourhood $U$ of 0, such that the only singularities of $f(U)\setminus\{0\}$ are transverse double points.

Notice that, in the real case, i.e. in the case of map germs $f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$, the converse implication of Theorem 2.1 does not hold. However, we have

Proposition 2.2**.**

If a map germ $f:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ is finitely determined then there exist a neighbourhood $U$ of 0 in $\mathbb{R}^{2}$ and a neighbourhood $V$ of 0 in $\mathbb{R}^{3}$, with $f(U)\subset V$, such that the only singularities of $f(U)\setminus\{0\}$ are transverse double points.

The following well-known theorem of Fukuda [12] will be also used in the next sections of the paper:

Theorem 2.3**.**

[12*]** *Suppose that $n\leq p$. Then given a semialgebraic subset $W$ of $J^{r}(n,p)$, there exist an integer $s\geq r$, depending only on $n,p$ and $r$, and a closed semialgebraic subset $\Sigma_{W}$ of $(\pi_{r}^{s})^{-1}(W)$, where $\pi_{r}^{s}:J^{s}(n,p)\to J^{r}(n,p)$ is the canonical projection, having codimension greater than 1 such that for any $C^{\infty}$-mapping $f:\mathbb{R}^{n}\to\mathbb{R}^{p}$ with $j^{s}f(0)$ belonging to $(\pi_{r}^{s})^{-1}(W)\setminus\Sigma_{W}$, there exists a positive number $\epsilon_{0}$ such that for any number $\epsilon$ with $0<\epsilon\leq\epsilon_{0}$ we have

1. (1)

$\tilde{S}_{\epsilon}^{n-1}=f^{-1}(S_{\epsilon}^{p-1})$ is a homotopy ($n-1$)-sphere which, if $n\neq 4,5$, is diffeomorphic to the natural ($n-1$)-sphere $S^{n-1}$, 2. (2)

the restricted mapping $f|_{\tilde{S}_{\epsilon}^{n-1}}:\tilde{S}_{\epsilon}^{n-1}\to{S}_{\epsilon}^{p-1}$ is topologically stable. Moreover, $f|_{\tilde{S}_{\epsilon}^{n-1}}$ is $C^{\infty}$-stable if ($n,p$) are nice dimensions, 3. (3)

denoting by $\tilde{B}_{\epsilon}^{n}=f^{-1}({B}_{\epsilon}^{p})$ the inverse image of the $p$-dimensional ball of radius $\epsilon$ centered at [math], the restricted mapping $f|_{\tilde{B}_{\epsilon}^{n}}:\tilde{B}_{\epsilon}^{n}\setminus\{0\}\to{B}_{\epsilon}^{p}\setminus\{0\}$ is proper, topologically stable ($C^{\infty}$ stable if $(n,p)$ are nice dimensions) and topologically equivalent ($C^{\infty}$ equivalent if $(n,p)$ are nice dimensions) to the product mapping

[TABLE]

defined by $(x,t)\mapsto(f(x),t)$ and 4. (4)

consequently, $f|_{\tilde{B}_{\epsilon}^{n}}:\tilde{B}_{\epsilon}^{n}\to{B}_{\epsilon}^{p}$ is topologically equivalent to the cone

[TABLE]

of the stable mapping $f|_{\tilde{S}_{\epsilon}^{n-1}}:\tilde{S}_{\epsilon}^{n-1}\to{S}_{\epsilon}^{p-1}$ defined by $C(f|_{\tilde{S}_{\epsilon}^{n-1}})(x,t)=(f(x),t).$

3. Some bi-Lipschitz Invariants

In this section, we consider the bi-Lipschitz equivalence in the ambient space. This means that two subsets $X,Y\subset\mathbb{K}^{p}$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, are bi-Lipschitz equivalent if there exists a homeomorphism in the ambient space $h:\mathbb{K}^{p}\to\mathbb{K}^{p}$ such that $h(X)=Y$.

Proposition 3.1**.**

Let $f,g:(\mathbb{K}^{n},0)\to(\mathbb{K}^{p},0)$ be two map germs, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ and $n\leq p$. If $f$ and $g$ are bi-Lipschitz equivalent, then

1. (1)

$D(f)$* and $D(g)$ are bi-Lipschitz equivalent.* 2. (2)

$\widetilde{D}(f)$* and $\widetilde{D}(g)$ are bi-Lipschitz equivalent.* 3. (3)

$f(D(f))$* and $g(D(g))$ are bi-Lipschitz equivalent.*

Proof.* * Assume that $f$ and $g$ are bi-Lipschitz equivalent, then there exist bi-Lipschitz homeomorphisms

[TABLE]

such that $f=k^{-1}\circ g\circ h.$

(1) We will prove that $h(D(f))=D(g)$. Take $x\in D(f)$, then there are two cases:

a) There exists $x^{\prime}\in\mathbb{K}^{n}$ such that $x\neq x^{\prime}$ and $f(x)=f(x^{\prime})$. Since $f=k^{-1}\circ g\circ h$, then $k\circ f=g\circ h$, one has

[TABLE]

Since $f(x)=f(x^{\prime})$, then $k(f(x))=k(f(x^{\prime}))$, hence $g(h(x))=g(h(x^{\prime}))$. Moreover, since $h$ is bijective, then $h(x)\neq h(x^{\prime})$. Consequently, $h(x)$ is a double point of $g$, or $h(x)\in D(g)$.

b) There exists a sequence $\{(x_{m},x^{\prime}_{m})\}\subset\mathbb{K}^{n}\times\mathbb{K}^{n}$ such that $x_{m}\neq x^{\prime}_{m}$ but $f(x_{m})=f(x^{\prime}_{m})$ and $x_{m}$ tends to $x$. We have $h(x)=\lim h(x_{m})$. In this case, with the same argument as above (with a remark that every bi-Lipschitz map is continous), we see that $(h(x),h(x))$ lies in the closure of the set $\{(x^{\prime},y^{\prime})\in\mathbb{K}^{n}\times\mathbb{K}^{n}:x^{\prime}\neq y^{\prime},g(x^{\prime})=g(y^{\prime})\},$ or $h(x)\in D(g)$.

We conclude that $h(D(f))\subset D(g).$

To prove that $D(g)\subset h(D(f))$ we proceed similarly, replacing $h$ by $h^{-1}$ in the previous argument.

(2) The fact “$\widetilde{D}(f)$ and $\widetilde{D}(g)$ are bi-Lipschitz equivalent” comes directly from (1). In fact, in this case $(h\times h)(\widetilde{D}(f))=\widetilde{D}(g)$, where $(h\times h)(x,y)=(h(x),h(y))$, with $(x,y)\in\widetilde{D}(f)$.

(3) We prove now that $f(D(f))$ and $g(D(g))$ are bi-Lipschitz equivalent. In fact, we prove that $k(f(D(f)))=g(D(g))$.

Take $y\in f(D(f))$, then there exists $x\in D(f)$ such that $y=f(x)$. Since $x\in D(f)$, then from the proof of (1), since $h(D(f))=D(g)$, hence $h(x)\in D(g)$. Since $k\circ f=g\circ h$ then $k(f(x))=g(h(x))$, therefore $k(y)=g(h(x))$. Consequently, $k(y)\in g(D(g))$ and hence $k(f(D(f)))\subset g(D(g))$. We proceed similarly to prove that $g(D(g))\subset k(f(D(f))),$ replacing $k$ by $k^{-1}.$

$\blacksquare$

From now, we assume that $p\geq 2n-1$. If $f$ is finitely determined and $p=2n-1,$ the image of the set of double points of $f$ is a curve, which can be embedded in the target space $\mathbb{K}^{2n-1};$ if $p>2n-1,$ the singularity of the image $X_{f}$ is isolated. [27, 28].

Proposition 3.2**.**

Let $f,g:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$ be two finitely determined map germs, where $p\geq 2n-1$. If $f$ and $g$ are bi-Lipschitz equivalent, then $link(f)$ and $link(g)$ are bi-Lipschitz equivalent.

Proof.* * Let $f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$, with $p\geq 2n-1$ be a finitely determined map germ, then there exists an open subset $U$ of [math] in $\mathbb{R}^{n}$ and an open subset $W$ of [math] in $\mathbb{R}^{p}$ such that

[TABLE]

is an immersion whose singularities are at most transverse double points. By Theorem 2.3, since $p\geq 2n-1$, then for $\epsilon$ enough small, $f^{-1}(S_{\epsilon}^{p-1})$ is homeomorphic to a sphere $S_{\epsilon}^{n-1}$ and

[TABLE]

is topologically stable. The same thing happens with $g$. Furthermore, in this case, $f|_{U\setminus\{0\}}$ is transverse to $S_{\epsilon}^{p-1}$. It follows that the inverse image $f^{-1}(S_{\epsilon}^{p-1})$ is diffeomorphism to the sphere $S_{\epsilon}^{n-1}$. Then, the images of the maps

[TABLE]

are respectively, the $link(f)$ and $link(g)$. Now, the bi-Lipschitz equivalence of $link(f)$ and $link(g)$ follows from the bi-Lipschitz equivalence of $f$ and $g$. $\blacksquare$

From Propositions 3.1 and 3.2, we have the following theorem:

Theorem 3.3**.**

Let $f,g:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$ be two finitely determined map germs, where $p\geq 2n-1$. If $f$ and $g$ are bi-Lipschitz equivalent, then $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and the images of the double point sets $f(D(f))$ and $g(D(g))$ are bi-Lipschitz equivalent.

4. Bi-Lipschitz classification in the case $(n,p)=(2,3)$

4.1. Hölder Complexes

We recall here some definitions which will be useful for our results later on.

There are two natural metrics defined on the spaces $X\subset\mathbb{R}^{n}$: the outer metric or euclidian metric

[TABLE]

which is the induced Euclidean metric on $X$ and the inner metric or length metric

[TABLE]

where $\Lambda(x,y)$ is the set of rectifiable arcs $\lambda:[0,1]\to X$ with $\lambda(0)=x$ and $\lambda(1)=y$ and $l(\lambda)$ is the length of $\lambda$. It is clear that the condition $d_{\rm out}(x,y)\leq d_{\rm in}(x,y)$ holds but the converse does not hold in general.

Definition 4.1**.**

We say that $X$ is Lipschitz normally embedded (LNE) if there exists $k>0$ such that for all $x,y\in X$, we have

[TABLE]

Notice that as the germ $f:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ is finitely determined, then we can consider $f$ as the germ of a polynomial map. Let $U$ be a neighbourhood of [math] in $\mathbb{R}^{2}$, then $f(U)$ is a semialgebraic surface of $\mathbb{R}^{3}$.

In this section, we consider the following equivalence relation between two semialgebraic sets:

Definition 4.2**.**

Let $X$ and $Y$ be two semialgebraic sets and $d_{X}$ and $d_{Y}$ be chosen metrics in $X$ and $Y$, respectively. We say that $X$ and $Y$ are abstract bi-Lipschitz equivalent if there exists a bi-Lipschitz homeomorphism $h:(X,d_{X})\to(Y,d_{Y})$ such that $h(X)=Y$. When the homeomorphism $h$ is semialgebraic, we say that $X$ and $Y$ are (abstract) semialgebraically bi-Lipschitz equivalent. **

When $d_{X}$ and $d_{Y}$ are the inner metrics, we say that $X$ and $Y$ are (abstract) bi-Lipschitz inner equivalent. Similarly, if $d_{X}$ and $d_{Y}$ are the outer metrics, we say that $X$ and $Y$ are (abstract) bi-Lipschitz outer equivalent. Notice that, in these cases, the homeomorphisms $h:(X,d_{X})\to(Y,d_{Y})$ is not necessarily defined in the whole ambient space.

Definition 4.3**.**

Let $A$ and $B$ be two arcs in $f(U)$, we define the contact order between $A$ and $B$ as

[TABLE]

where $S(0,r)$ is the sphere of center [math] and radius $r$ in $\mathbb{R}^{3}$.**

Lev Birbrair [2] introduced a construction called Hölder Complex in order to study bi-Lipschitz classification of semialgebraic surfaces. We provide some definitions and results of this construction. Notice that this study is performed on the surfaces, and the metric is the inner metric. Birbrair’s construction is summarized as follows:

Let $\Gamma$ be a finite graph. We denote by $E_{\Gamma}$ the set of edges and by $V_{\Gamma}$ be the set of vertices of $\Gamma$.

Definitions 4.4**.**

A Hölder Complex is a pair $(\Gamma,\beta)$, where $\beta:E_{\Gamma}\to\mathbb{Q}$ such that $\beta(g)\geq 1$, for every $g\in E_{\Gamma}$.

Two Hölder Complexes $(\Gamma_{1},\beta_{1})$ and $(\Gamma_{2},\beta_{2})$ are called combinatorially equivalent if there exists a graph isomorphism $i:\Gamma_{1}\to\Gamma_{2}$ such that, for every $g\in E_{\Gamma_{1}}$, we have $\beta_{2}(i(g))=\beta_{1}(g)$.

The standard $\beta$-Hölder triangle $T_{\beta}$, with $\beta\in[1,\infty[\,\cap\,\mathbb{Q}$, is the semialgebraic subset of $\mathbb{R}^{2}$ defined by

[TABLE]

Definition 4.5**.**

A (semialgebraic) subset $X$ of $\mathbb{R}^{3}$ is called a $\beta$-Hölder triangle with the principal vertex $a\in X$ if the germ $(X,a)$ is (semialgebraically) outer bi-Lipschitz equivalent to the germ $(T_{\beta},0)$. **

** Remark 4.6****.**

Let $X$ be a $\beta$-Hölder triangle then the contact order between the two branches $\gamma_{1}$ and $\gamma_{2}$ is equal to $\beta$ (see Figure 1).**

Theorem 4.7** (Theorem 4.1, [4]).**

Let $(X,0)$ and $(Y,0)$ be semialgebraic curves with branches $X_{1},\ldots,X_{l}$ and $Y_{1},\ldots,Y_{s}$. Then $(X,0)$ is outer bi-Lipschitz equivalent to $(Y,0)$ if and only if $l=s$ and there is a permutation $\sigma$ of $\{1,\ldots,l\}$ such that

[TABLE]

Definition 4.8**.**

Let $(\Gamma,\beta)$ be a Hölder Complex. A set $X\subset\mathbb{R}^{3}$ is called a (semialgebraic) Geometric Hölder Complex corresponding to $(\Gamma,\beta)$ with the principal vertex $a\in X$ if:

1. (1)

There exists a homeomorphism $\psi:C\Gamma\to X$, where $C\Gamma$ is the topological cone over $\Gamma$. 2. (2)

Let $\alpha\subset C\Gamma$ be the vertex of $C\Gamma$, then $\psi(\alpha)=a$. 3. (3)

For each $g\in E_{\Gamma}$, the set $\psi(Cg)$ is a (semialgebraic) $\beta(g)$-Hölder triangle with the principal vertex $a$, where $Cg\subset C\Gamma$ is the subcone over $g$.

Theorem 4.9** (Theorem 6.1, [2]).**

Let $X\subset\mathbb{R}^{3}$ be a two-dimensional closed semialgebraic set and let $a\in X$. Then there exist a number $\delta>0$ and a Hölder Complex $(\Gamma,\beta)$ such that $B(a,\delta)\cap X$ is a semialgebraic Geometric Hölder Complex corresponding to $(\Gamma,\beta)$ with the principal vertex $a$, where $B(a,\delta)$ is the closed ball centered at $a$ of radius $\delta$.

Definition 4.10**.**

We say that $b$ is a non-critical vertex of $\Gamma$ if it is incident with exactly two different edges $g_{1}$ and $g_{2}$ and these edges connect two different vertices $b_{1}$ and $b_{2}$ with $b$.

If this vertex $b$ is connected by $g_{1}$ and $g_{2}$ with only one other vertex $b^{\prime}$, we say that $b$ is a loop vertex.

The other vertices of $\Gamma$ (which are neither non-critical nor loop) are called critical vertices of $\Gamma$. **

Given a Geometric Hölder Complex, there is a simplification process described by Lev Birbrair (Theorem 7.3, [2]), allowing to assume that every vertex in $V_{\Gamma}$ is a either a critical vertex or a loop vertex. The resulting Geometric Hölder Complex is called a Canonical Hölder Complex of $X$ at $a$. Two simplifications of the same Hölder Complex are combinatorially equivalent.

Theorem 4.11** (Birbrair Classification Theorem, Theorem 8.1 [2]).**

Let $X_{1},X_{2}\subset\mathbb{R}^{3}$ be two dimensional semialgebraic subsets with $a_{1}\in X_{1}$ and $a_{2}\in X_{2}$. The germs $(X_{1},a_{1})$ and $(X_{2},a_{2})$ are (abstract) bi-Lipschitz inner equivalent if and only if the Canonical Hölder Complexes of $X_{1}$ at $a_{1}$ and $X_{2}$ at $a_{2}$ are combinatorially equivalent.

Notice that the original Classification Theorem in [2] holds for surfaces in $\mathbb{R}^{n}$, for $n\geq 3$.

4.2. Bi-Lipschitz classification in the case $(n,p)=(2,3)$.

Let $f,g:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ be map germs. From now, we denote by $X_{f}=f(U)$ and $X_{g}=g(U)$, where $U$ is a sufficiently small neighbourhood of [math]. As an application of Birbrair’s Theorem, we have the following proposition:

Proposition 4.12**.**

Let $f,g:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ be finitely determined map germs. Then $X_{f}$ and $X_{g}$ are (abstract) bi-Lipschitz inner equivalent if and only if $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and $f(D(f))$ and $g(D(g))$ are bi-Lipschitz equivalent.

Proof.* *

Assume that $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and $f(D(f))$ and $g(D(g))$ are bi-Lipschitz equivalent.

As $f,g:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ are finitely determined map germs, one can consider $f$ and $g$ as polynomial maps. Then $X_{f}$ and $X_{g}$ are closed semialgebraic surfaces. By Theorem 4.9, there exists a semialgebraic Hölder Complex $(\Gamma_{f},\beta)$ such that $X_{f}$ is a semialgebraic Geometric Hölder Complex corresponding to $(\Gamma_{f},\beta)$ with the principal vertex [math]. Similar property holds for $X_{g}.$

By Birbrair’s simplification process, the germs $X_{f}$ and $X_{g}$ correspond to two Canonical Hölder Complexes at 0, that we denote by $\tilde{\Gamma}_{f}$ and $\tilde{\Gamma}_{g}$.

Since the links $link(f)$ and $link(g)$ are bi-Lipschitz equivalent, the graphs $\Gamma_{f}$ and $\Gamma_{g}$ are homeomorphic. Then the loop vertices and the critical vertices of $\Gamma_{f}$ correspond respectively to the loop vertices and the critical vertices of $\Gamma_{g}$.

Notice that the critical vertices of $\Gamma_{f}$ are precisely the intersection points of $link(f)$ and $f(D(f)).$ We have the same thing for $\Gamma_{g}$. Then the branches of $f(D(f))$ correspond to the branches of $g(D(g))$.

By Theorem 4.7, since $f(D(f))$ and $g(D(g))$ are bi-Lipschitz equivalent then the contact order $\beta$ of two branches of $f(D(f))$ is the same the contact order of the two corresponding branches of $g(D(g))$.

Hence, the two Canonical Hölder Complexes $\tilde{\Gamma}_{f}$ and $\tilde{\Gamma}_{g}$ are combinatorially equivalent. The proof of the Proposition follows from Birbrair’s Classification Theorem 4.11.

For the converse, let us suppose that $X_{f}$ and $X_{g}$ are semialgebraically bi-Lipschitz equivalent with respect to the inner metric. From the Classification Theorem, it follows that the associated Hölder Complexes $\tilde{\Gamma}_{f}$ and $\tilde{\Gamma}_{g}$ are combinatorially equivalent. As a consequence, we get the outer bi-Lipschitz equivalence between $f(D(f))$ and $g(D(g)).$ Notice that this bi-Lipschitz equivalence can be extended to the ambient space (see [4]).

Furthermore, as $f$ and $g$ are finitely determined, the only singularities of $link(f)$ and $link(g)$ are transverse double points, then these two sets are Lipschitz normally embedded. Hence, it follows that $link(f)$ and $link(g)$ are outer bi-Lipschitz equivalent.

$\blacksquare$

Corollary 4.13**.**

Let $f,g:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ be finitely determined map germs such that $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and $f(D(f))$ and $g(D(g))$ are bi-Lipschitz equivalent. If $X_{f}$ and $X_{g}$ are LNE then $X_{f}$ and $X_{g}$ are bi-Lipschitz equivalent with respect to the outer metric.

5. Surfaces with homogeneous parametrization

Let $f,g:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ be finitely determined map germs of corank 1 with homogeneous parametrization of the same degrees, that means that $f$ and $g$ can be written in the following way

[TABLE]

where $f_{j}$ and $g_{j}$ are homogeneous polynomials of the same degree $d_{j}$, for $j=2,3.$

Notice that the equation $\phi(x,y,z)=0$ defining $X_{f}$ (resp., $X_{g}$) is not homogeneous in general.

The following theorem shows that in this case, in order to obtain the bi-Lipschitz equivalence of $X_{f}$ and $X_{g}$, we do not need any hypothesis on the bi-Lipschitz equivalence of the images of the double point sets.

Theorem 5.2**.**

Let $f,g:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ be finitely determined map germs of corank 1 with homogeneous parametrization of the form (5.1). If $link(f)$ and $link(g)$ are bi-Lipschitz equivalent then $X_{f}$ and $X_{g}$ are bi-Lipschitz equivalent with respect to the inner metric.

Proof.* * Let us write $f$ in the form (5.1), that means

[TABLE]

Acording to [14], the set of double points $D(f)$ is the set of the solutions of the following system:

[TABLE]

Then $D(f)$ is the union of lines $\Gamma_{i}$ whose equations are of the type $y=\alpha_{i}x$ and in some cases also the line $x=0$. In the case $y=\alpha_{i}x$, we have

[TABLE]

We see that $f_{2}(x,\alpha_{i}x)$ is a homogeneous polynomial of degree $d_{2}$ and $f_{3}(x,\alpha_{i}x)$ is a homogeneous polynomial of degree $d_{3}$ with respect to the variable $x$. Then the image of the set of double points $f(D(f))$ is the union of parametrized curves of the form $(x,\tilde{f}_{2},\tilde{f}_{3})$ where $\tilde{f}_{j}$ are homogeneous polynomials of degree $d_{j}$ with respect to variable $x$, for $j=2,3$. We have the same thing for $g$, that is $g(D(g))$ is the union of parametrized curves of the form $(x,\tilde{g}_{2},\tilde{g}_{3})$ where $\tilde{g}_{j}$ are homogeneous polynomials of the same degree than $\tilde{f}_{j}$, for $j=2,3$. Therefore, acording to Theorem 4.7, the order of contact of two branches of $f(D(f))$ is the same than the order of contact of two corresponding branches of $g(D(g))$.

In the case that $x=0$ is a branch of $D(f)$ and $D(g)$, it provides two corresponding branches respectively of $f(D(f))$ and $g(D(g))$ (see Proposition 6.2, [24]).

Moreover, with the hypothesis that $link(f)$ and $link(g)$ are bi-Lipschitz equivalent then the graphs of the Hölder complexes of $X_{f}$ and $X_{g}$ are equivalent. Therefore, the Canonical Hölder Complexes of $X_{f}$ and $X_{g}$ are combinatorially equivalent. By the Classification Theorem 4.11, it follows that $X_{f}$ and $X_{g}$ are bi-Lipschitz equivalent with respect to the inner metric. $\blacksquare$

Corollary 5.3**.**

Let $f,g:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ be finitely determined map germs of corank 1 with homogeneous parametrization. If $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and $X_{f}$ and $X_{g}$ are LNE, then $X_{f}$ and $X_{g}$ are bi-Lipschitz equivalent with respect to the outer metric.

In the next proposition we prove that in the space of corank 1 finitely determined map-germs with homogeneous parametrization, the topological classification of the map-germs and the inner bi-Lipschitz classification of their images coincide.

Proposition 5.4**.**

Let $H_{n,m}$ be the space of real polynomial mappings

[TABLE]

where $p(x,y)=\sum^{n}_{i=0}a_{i}x^{n-i}y^{i}\,\,\,and\,\,q(x,y)=\sum^{m}_{i=0}b_{i}x^{m-i}y^{i},$ with $\text{gcd}(n,m)=1.$ Then

• (1)

The subset $F_{n,m}:=\{f\in H_{n,m}\,|\,\,f\,\,\text{is}\,\,\mathcal{A}$-$\text{finitely determined}\,\}$* is a non empty open and dense subset of $H_{n,m}.$*

• (2)

If $f,g\in F_{n,m},$ then $f$ and $g$ are ${\mathcal{C}^{0}}$-$\mathcal{A}$-equivalent if and only if $X_{f}$ and $X_{g}$ are bi-Lipschitz equivalent.

Proof.* * We first notice that $F_{n,m}\neq\emptyset$ since $f(x,y)=(x,y^{n},(x+y)^{m}),$ where $gcd(n,m)=1$, is finitely determined (see [23], Proposition 9.8). Then $F_{n,m}$ is an open and dense subset of $H_{n,m}$ follows because finite determinacy is an open condition ([8]) and its complement is a proper semialgebraic set in $H_{n,m}$ (see [23] and [24], Example 5.5 and Lemma 7.1). Moreover, on each connected component of $F_{n,m}$, the $C^{0}$-$\mathcal{A}$ type is constant (see Damon’s Theorem 1 in [8], page 381). It follows from Corollary 3.4 in [15] and Theorem 5.2 that:

$f$ and $g$ are ${\mathcal{C}^{0}}$-$\mathcal{A}$-equivalent if and only if $X_{f}$ and $X_{g}$ are bi-Lipschitz equivalent.

$\blacksquare$

6. Homogeneous surfaces

Theorem 5.2 provides a class of quasi-homogeneous surfaces for which the bi-Lipschitz type of the link determines the bi-Lipschitz type of the surface relatively to the inner metric. However, we do not know if the result is true for the more general case of quasi-homogeneous surfaces, with non-isolated singularities, which do not admit homogeneous parametrization. The study of the topological type of surfaces with non-isolated singularities is an important problem and has been investigated both in real and complex cases. This study is interesting even for the homogeneous case, for instance, see [10]. We will prove the following theorem which shows that the result is true in the homogeneous case. Moreover, in this case, the result is true for the outer metric without the condition that the surface is Lipschitz normally embedded.

Theorem 6.1**.**

Let $X\subset(\mathbb{R}^{3},0)$ and $Y\subset(\mathbb{R}^{3},0)$ be two map germs such that $X=\phi^{-1}(0)$ and $Y=\phi^{\prime-1}(0)$, where $\phi$ and $\phi^{\prime}:\mathbb{R}^{3}\to\mathbb{R}$ are two homogeneous polynomials. Assume that $link(X)$ and $link(Y)$ are bi-Lipschitz equivalent, then $X$ and $Y$ are bi-Lipschitz equivalent with respect to the outer metric.

Proof.* * Recall that a homogeneous polynomial $\phi$ has the form $\sum_{i\geq 1}\lambda_{i}x_{1}^{\alpha_{i}}x_{2}^{\beta_{i}}x_{3}^{\gamma_{i}}$ where $\alpha_{i}+\beta_{i}+\gamma_{i}$ is a positive integer constant, for every $i\geq 1$. Equivalently, there is the action of $\mathbb{R}^{*}_{+}$ on $X$ defined by:

[TABLE]

The orbit of a point $p\in X$ under this action is a half-line from [math], containing $p$, denoted by $O(p)$. The same thing happens for $Y$. Notice that, since $link(X)$ and $link(Y)$ are bi-Lipschitz equivalent, then there exists a bi-Lipschitz diffeomorphism $H_{L}:link(X)\to link(Y)$ which sends the 2-dimensional sphere $S^{2}_{1}$ of radius $1$ centered at [math] in $\mathbb{R}^{3}$ to the 2-dimensional sphere $S^{2}_{R}$ of radius $R$ centered at 0 in $\mathbb{R}^{3}$ (cf. Theorem 2.3).

Let us define a map $H:X\to Y$ as follows: let $p$ be a point of $X$, denote by $S^{2}_{\epsilon}$ the 2-dimensional sphere of radius $\epsilon$, centered at [math] and containing $p$. The point $p^{\prime}=S_{1}^{2}\cap O(p)$ belongs to $link(X)$.

The point $H_{L}(p^{\prime})$ belongs to $Y$ as well as its orbit $O(H_{L}(p^{\prime}))$ under the action (6.2), but for $Y$. Notice that $H_{L}(p^{\prime})\in S^{2}_{R}$ (see figure 3). Let us define

[TABLE]

It is clear that $H$ is bijective.

In the same way, take another point $q\in X$ and denote $\epsilon^{\prime}$ such that $q\in S_{\epsilon^{\prime}}^{2}$. Let us denote $q^{\prime}=S_{1}^{2}\cap O(q)$, we have

[TABLE]

(see Figure 4).

We will prove that $d(H(p),H(q))\leq C\;d(p,q)$ where $C$ is a positive real constant and $d$ is the Euclidean distance (“outer” distance [5]) in the respective Euclidean spaces.

Let us define $w=O(H_{L}(p^{\prime}))\cap S_{\epsilon^{\prime}}^{2}$ and $v=H^{-1}(w)$, then $v\in S_{\epsilon^{\prime}}^{2}$. We have

[TABLE]

1. At first, we approximate $d(w,H(q))$ to $d(p,q)$. On the one hand, we have $p=(a_{1},a_{2},a_{3})\in S^{2}_{\epsilon}$, then $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=\epsilon^{2}$. On the other hand, $p^{\prime}\in O(p)\cap S^{2}_{1}$ and $p^{\prime}=(ta_{1},ta_{2},ta_{3})$, where $t\in\mathbb{R}$, hence one has $t=1/\epsilon$. Consequently, we have

[TABLE]

Since $H_{L}(p^{\prime})\in S^{2}_{R}$, then if we denote the coordinates of $H_{L}(p^{\prime})$ by $H_{L}(p^{\prime})=(b_{1},b_{2},b_{3})$, we have $b_{1}^{2}+b_{2}^{2}+b_{3}^{2}=R^{2}$. In the same way as above, we have

[TABLE]

Similarly, if $q=(x_{1},x_{2},x_{3})$ and $H_{L}(q^{\prime})=(y_{1},y_{2},y_{3})$, then by an easy calculus, we have

[TABLE]

and

[TABLE]

Therefore, we have

[TABLE]

where $C_{H_{L}}$ is the bi-Lipschitz constant of the bi-Lipschitz homeomorphism $H_{L}$.

Now, notice that we can choose the point $p=(a_{1},a_{2},a_{3})$ such that $a_{1},a_{2}$ and $a_{3}$ are non-negative. Moreover, we can assume that $\epsilon\leq\epsilon^{\prime}$, and then, from (6.3) and (6.4), we have

[TABLE]

From (6.5) and (6.6), we have

[TABLE]

2. We prove now that $d(H(p),w)\leq d(p,q)$. We have $d(H(p),w)=d(p,v)=\epsilon^{\prime}-\epsilon$ (see Figure 4). The problem reduces to prove that $d(p,v)\leq d(p,q)$. We consider the triangle $pvq$ in the plane $(P)$ defined by the three points $0,p$ and $q$. An easy computation (see Figure 5) shows that the angle $pvq$ is bigger that the angle $pqv$, that we denote by

[TABLE]

Let $s$ and $t$ be the intersections of $qp$ and $q0$ with the sphere $S_{\epsilon^{\prime}}^{2}$, respectively. Notice that when $p$ is fixed, the point $s$ lays always between the two points $t$ and $v$ on the circle $S_{\epsilon^{\prime}}^{2}\cap(P)$, then the inequality (6.7) always happens. Futhermore, $\angle pqv=\angle pvq$ when $0,p$ and $q$ are colinear and $p$ lays on the segment $[0,q]$.

From (6.5) and (6.7), we have

[TABLE]

where $C={\rm max}\{1+\frac{C_{H_{L}}}{R},2\}$. Thus, the map $H$ is a bi-Lipschitz homeomorphism and $X$ is bi-Lipschitz equivalent to $Y$ with respect to the outer metric.

$\blacksquare$

Acknowledgement: We thank Otoniel Nogueira da Silva for many useful conversations. The first named author thanks the Project UNESP-FAPESP no 2015/06697-9, the Project USP-COFECUB no UC Ma 163-17 and the Project USP-FAPESP no 2018/07040-1. The second and third named authors were partially supported by FAPESP, grant $\#2014/00304-2$ and CNPq, grant $\#433257/2016-4.$ The third author was partially supported by CNPq, grant $\#306306/2015-8.$